Once the rotation matrix has been estimated, the translation vector t can be estimated as:

The ICP algorithm is iterative because it iteratively improves the tentative correspondences. If true correspondences were known, clearly the process could operate in one pass. ICP has two main steps in its inner loop: (i) closest point computation and (ii) rigid transformation estimation. In more detail, the algorithm operates as follows:

1.For each data-point di D, compute the closest point mj M according to Eq. (6.3).

2.With the correspondences (di , mj ) from step 1, estimate the new transformation parameters a = (R, t).

3.Apply the new transformation parameters a from step 2 to the point cloud D.

4.If the change in EICP(a, D, M) between two successive iterations is lower than a threshold then terminate, else go to step 1.

It was proven [7] that this algorithm is guaranteed to converge monotonically to a local minimum of Eq. (6.2). Note that, as for any local iterative method, a strategy for initializing a must be used. An overview of the most popular initialization strategies is given in Sect. 6.2.3.1.

6.2.3 ICP Extensions

Although ICP has been successfully applied to many registration problems, there are several critical issues that need to be taken care of. In particular, ICP performs well when the following assumptions are met:

1.The two views must be close to each other. If not, ICP will probably get stuck in a local minimum. This issue is typically solved by pre-alignment of the two 3D views, also called coarse registration.

2.The two views must fully overlap or the data-view D must be a subset of the model-view M. The problem arises from the fact that ICP always assigns a closest model point to every data point. If a data point has no corresponding model point, this will create a spurious correspondence, an outlier with respect to the sought transformation, that will bias the solution or prevent the algorithm from finding the correct transformation parameters.

Two other important issues are the speed of computation and the accuracy of the ICP algorithm. Typically, methods focused on speed improvement try to speed up the closest point computation step, which is the bottleneck of the algorithm. Other interesting approaches address the speed of convergence by proposing new distance formulations for the problem described by Eq. (6.1). Methods focusing on accuracy exploit additional information in order to measure the similarity between corresponding points not only in terms of proximity. In the following, we describe